Global Carleman inequalities for parabolic systems and applications to controllability

SIAM J. CONTROL OPTIM.c 2006 Society for Industrial and Applied Mathematics Vol. 45, No. 4, pp. 1395–1446 GLOBAL CARLEMAN INEQUALITIES FOR PARABOLIC SYSTEMS AND APPLICATIONS TO CONTROLLABILITY∗ ENRIQUE FERN´ ANDEZ-CARA†AND SERGIO GUERRERO† Abstract. This paper has been conceived as an overview on the controllability properties of some relevant (linear and nonlinear) parabolic systems. Specifically, we deal with the null controllability and the exact controllability to the trajectories. We try to explain the role played by the observability inequalities in this context and the need of global Carleman estimates. We also recall the main ideas used to overcome the difficulties motivated by nonlinearities. First, we considered the classical heat equation with Dirichlet conditions and distributed controls. Then we analyze recent extensions to other linear and semilinear parabolic systems and/or boundary controls. Finally, we review the controllability properties for the Stokes and Navier–Stokes equations that are known to date. In this context, we have paid special attention to obtaining the necessary Carleman estimates. Some open questions are mentioned throughout the paper. We hope that this unified presentation will be useful for those researchers interested in the field. Key words. Carleman inequalities, parabolic equations, unique continuation, observability, null controllability AMS subject classifications. 35B37, 70Q05, 93B05, 93B07 DOI. 10.1137/S0363012904439696 Introduction. The goal of this paper is to provide a panorama of an important subfield of control theory: the null controllability analysis of parabolic equations and systems. The main contributions to this area are due to O. Yu. Imanuvilov, who popularized the use of global Carleman estimates in the context of null controllability. Many arguments from [36], [30], [39], and [37] will be reproduced here. Another relevant contributor has been E. Zuazua, who was able to deduce global controllability results for some nonlinear systems for the first time in [56]. The controllability of partial differential equations has been the object of intensive research during the last few decades. In 1978, Russell [53] made a survey of the most relevant results that were available in the literature at that time. In that paper, the author described a number of different tools that were developed to address controllability problems, often inspired and related to other subjects concerning partial differential equations: multipliers, moment problems, nonharmonic Fourier series, etc. More recently, J.-L. Lions introduced the so-called Hilbert uniqueness method (for instance, see [44], [45], [46]). That was the starting point of a fruitful period on the subject. It would be impossible to present here all the relevant results that have been proved in this area. We will thus reduce our scope drastically, considering only null controllability problems for parabolic equations and systems. In order to get an idea, let us consider the simplest case of the linear heat equation ∗Received by the editors January 14, 2004; accepted for publication (in revised form) April 10, 2006; published electronically October 12, 2006. This work was partially supported by D.G.E.S. (Spain), grant BFM2003-06446. http://www.siam.org/journals/sicon/45-4/43969.html †Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain ([email protected], [email protected]). 1395 Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1396 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO with Dirichlet boundary conditions and distributed controls: ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy=v1Oin Q=Ω×(0,T), y=0 on Σ=∂Ω×(0,T), y(0) = y0in Ω. (0.1) Here, Ω ⊂RNis a bounded domain of class C2,O⊂Ω is a nonempty open subset, 1Ois the characteristic function of O, and Tis a given positive time. We assume that the initial state y0is given in L2(Ω) and try to find a control v∈L2(O×(0,T)) such that the associated state y=y(x, t) possesses a desired behavior at time t=T. Roughly speaking, systems of parabolic type like (0.1) are characterized by nonreversibility, the dissipativity of the solutions (i.e., the fact that energy is lost along the trajectories), and the regularizing effect. In accordance with this last property, it is not possible to lead the solutions of (0.1) exactly to every final state in a Sobolev space like L2(Ω) or Hm(Ω), unless the control is exerted brutally, i.e., unless we are in the trivial and uninteresting situation O=Ω. On the contrary, it may be interesting to investigate whether it is possible to drive the solutions to (0.1) (acting on a small set O⊂⊂Ω) exactly to a state on a trajectory. For instance, it is completely meaningful to search for controls vsuch that the associated states ysatisfy y(T)=0 in Ω.(0.2) If such controls exist, we say that (0.1) is null controllable at time T. Observe that the null controllability of a parabolic system is a very useful property from the viewpoint of applications. Indeed, it permits us to reach in a finite amount of time a state that is “natural” for the system. All the work has to be done for t∈[0,T]. Afterwards, no additional effort is needed to get a satisfactory situation. It is also easy to see that, for a linear system of the kind above, null controllability is equivalent to exact controllability to the trajectories, i.e., to the following property: for any y0∈L2(Ω) and any function ysatisfying yt−Δy=0 in Q, y= 0 on Σ and y∈C0([0,T]; L2(Ω)), there exist controls vsuch that y(T)=y(T)inΩ. It will be explained below that the null controllability of a linear parabolic system is, roughly speaking, equivalent to the observability of the associated adjoint states. More precisely, for each ϕ0∈L2(Ω), let us consider the so-called adjoint system ⎧ ⎪ ⎨ ⎪ ⎩ −ϕt−Δϕ=0 in Q, ϕ= 0 on Σ, ϕ(T)=ϕ0in Ω. (0.3) Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1397 Then (0.1) is null controllable with controls in L2(O×(0,T)) if and only if there exists C>0 such that ϕ(0)2 L2(Ω) ≤CO×(0,T )|ϕ|2dx dt ∀ϕ0∈L2(Ω).(0.4) At present, the most powerful tools to prove inequalities like (0.4) for general parabolic systems are global Carleman estimates. They have the form Ω×(0,T ) ρ2|ϕ|2dx dt ≤CO×(0,T ) ρ2|ϕ|2dx dt,(0.5) where ρ=ρ(x, t) is continuous and strictly positive for t∈(0,T). The previous considerations motivate the need for establishing such estimates. It is also natural to ask if null controllability and/or exact controllability to the trajectories hold in the framework of semilinear or even genuinely nonlinear parabolic systems. For instance, we can be interested in controls and states satisfying ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy+f(y)=v1Oin Q, y= 0 on Σ, y(0) = y0in Ω (0.6) and (0.2), where the function f:R→ Ris given and, for instance, we assume that f∈C1(R) and f(0)=0. For systems of this kind, a classical approach relies on the use of fixed point arguments. The idea is to find a fixed point of the mapping z→ y, where yis, together with some v, a solution to the linearized system ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ yt−Δy+f(z) zy=v1Oin Q, y= 0 on Σ, y(0) = y0in Ω (0.7) satisfying (0.2). The existence of vis again equivalent to an observability inequality which is implied by appropriate Carleman estimates similar to (0.5). The fixed point approach can be applied whenever, among other things, the way the constants arising in Carleman estimates depend on the coefficients of the linearized systems is known in detail. Therefore, a careful analysis of (0.5) is needed. The theoretical results we present below can be applied in many contexts. For instance, they can serve as tools for •controlling the temperature of a medium eventually influenced by transport and/or chemical effects; •controlling the velocity field of a real fluid (like those modeled by the Navier– Stokes equations), etc. In the following sections, we will explain how these ideas can be applied to several parabolic equations and systems. A similar (but different) overview on this topic is [6], where the author is also concerned with the stabilization of parabolic systems. An important aspect of the controllability of time-dependent systems that will not be treated in this paper is the analysis and obtainment of numerical approximations. Of course, the most interesting question concerning the null controllability of (0.1) and (0.6) is how to construct a control with the desirable properties. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1398 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO Several ideas that reproduce the theoretical arguments at a finite-dimensional level were given, for instance, in [34]. Unfortunately, the proposed algorithms seem to be still too expensive. We simply point out here that the controllability analysis we recall in section 2, based on the solution of a linear fourth order problem, could provide some promising ideas. There are many other important subjects related to the controllability of parabolic equations that will not be visited here. Let us mention some of them: •The analysis of the existence of insensitizing controls in the sense of [47] or, more generally, the null controllability of cascade systems. This has been the objective of considerable work in recent years (see, for instance, [12], [9], and [26]). •The controllability of parabolic systems with memory. This includes interesting families of linear viscoelastic fluids. Some partial results in this field are given in [7] and [14]. •The control of coupled systems of parabolic-hyperbolic type, like a combination of heat and wave equations, the system of thermoelasticity, etc. Some of these problems have been treated in [57], [42], and [43]. •The control of fluid-solid interaction systems. This seems to be unexplored at present. For very simple models, some results are given in [13]. •The control of nondeterministic systems. Some advances were made in the context of linear problems. See, for instance, [8]. In this field, the main open question turns out to be the obtainment of controllability results for nonlinear systems, due to the lack of compactness. The plan of this paper is the following. In section 1, we consider the classical heat equation and some variants. Essentially, we deal there with heat equations with strong solutions. We recall the proof of a basic Carleman estimate and present some applications to semilinear systems. Section 2 is devoted to the null controllability analysis of other more complicated parabolic equations involving weak and/or very weak solutions. It will be shown that the proof of the related Carleman inequalities is much more involved and needs some extra work. We will also indicate how this can be used to deduce the exact controllability to the trajectories of more general semilinear systems. Finally, section 3 is concerned with systems of the Stokes kind. Here, the main difficulties are caused by the pressure. We will recall a recent argument leading to an appropriate but not completely satisfactory Carleman inequality. We will also explain how this leads to the null controllability of Stokes-like systems and to the local exact controllability to the trajectories of the Navier–Stokes equations. 1. Null controllability of the linear heat equation and applications. 1.1. Null controllability and observability. Let us consider again the simplest case of the linear heat equation with Dirichlet boundary conditions and distributed control with support in a small set: ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy=v1Oin Q, y= 0 on Σ, y(0) = y0in Ω. (1.1) We assume that O⊂⊂Ω is a nonempty (small) open subset and Tis a given positive time. In what follows, n(x) will stand for the outward unit normal vector at the point x∈∂Ω. We assume that the initial state y0is given in L2(Ω), and we try to Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1399 find a control v∈L2(O×(0,T)) such that the associated state y=y(x, t) possesses a desired behavior at time t=T. Recall that under these assumptions, system (1.1) has a unique weak solution y satisfying y∈L2(0,T;H1 0(Ω)) ∩C0([0,T]; L2(Ω)) that depends continuously on y0and v. Our interest is to provide an answer to the following questions. Question 1: Null controllability. For every y0∈L2(Ω), can a control v∈L2(O× (0,T)) be found such that y(T) = 0 in Ω? This question will be answered affirmatively in this section. To this end, we will introduce, for each ϕ0∈L2(Ω), the adjoint system ⎧ ⎪ ⎨ ⎪ ⎩ −ϕt−Δϕ=0 in Q, ϕ= 0 on Σ, ϕ(T)=ϕ0in Ω, (1.2) and we will try to answer the following auxiliary question. Question 2: Observability inequality. Can we find a constant C>0 such that, for each ϕ0∈L2(Ω), the associated solution of (1.2) satisfies ϕ(0)2 L2(Ω) ≤CO×(0,T )|ϕ|2dx dt ?(1.3) An affirmative answer to Question 2 implies an affirmative answer to Question 1. This is what is proved in the following result. Theorem 1.1. The observability inequality (1.3) implies the null controllability of (1.1). Proof. We divide the proof into two steps. First, we build a sequence of controls vε∈L2(O×(0,T)) with ε>0 which provide the approximate controllability of (1.1) (see (1.7)). Second, we pass to the limit when εtends to zero and we conclude. Step 1. Let y0∈L2(Ω) and ε>0 be given. Let us introduce the functional Jε, with Jε(ϕ0)=1 2O×(0,T )|ϕ|2dx dt +εϕ0L2(Ω) +(ϕ(0),y0)L2(Ω) (1.4) for every ϕ0∈L2(Ω). Here, ϕis the solution of (1.2) associated to the initial condition ϕ0. Using (1.3), it is not difficult to check that Jεis strictly convex, continuous, and coercive in L2(Ω), so it possesses a unique minimum ϕ0 ε∈L2(Ω), whose associated solution is denoted by ϕε. Let us now introduce the control vε=ϕε1O, and let us denote by yεthe solution of (1.1) associated to vε. Let y1be the final state of the solution to (1.1) with vanishing control. Let us remark that the unique interesting case to be studied turns out to be when y1L2(Ω) > ε, since this is equivalent to ϕ0 ε= 0. See [19] for more details. Under this assumption, we can differentiate the functional Jεat ϕ0 εand obtain a necessary condition for Jε to reach a minimum at ϕ0 ε,say, O×(0,T ) ϕεϕdxdt+εϕ0 ε ϕ0 εL2(Ω) ,ϕ 0L2(Ω) +(ϕ(0),y0)L2(Ω) =0(1.5) for every ϕ0∈L2(Ω). Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1400 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO Using this and (1.3) for ϕ0=ϕ0 ε, we obtain vεL2(O×(0,T )) ≤√Cy0L2(Ω), where Cis the observability constant of (1.3). Since systems (1.1) and (1.2) are in duality, we have O×(0,T ) ϕεϕdxdt=(yε(T),ϕ 0)L2(Ω) −(y0,ϕ(0))L2(Ω),(1.6) which, combined with (1.5), yields yε(T)L2(Ω) ≤ε.(1.7) Step 2. Since the sequence {vε}is bounded in L2(O×(0,T)), it possesses a (weakly) convergent subsequence to certain v∈L2(O×(0,T)). Using classical parabolic estimates we deduce that, at least for a subsequence, yε→yweakly in L2(0,T;H1 0(Ω)) ∩H1(0,T;H−1(Ω)),(1.8) where yis the solution of (1.1) with control v. In particular, this gives weak convergence for {yε(t)}(t∈[0,T]) in L2(Ω) so we have y(T)=0. Remark 1. At this point, some comments must be made: 1. We have proved that (1.3) implies null controllability with a control that satisfies vL2(O×(0,T )) ≤√Cy0L2(Ω),(1.9) where Cis the observability constant. Conversely, if we have null controllability with controls v∈L2(O×(0,T)) that satisfy vL2(O×(0,T )) ≤√Cy0L2(Ω) (1.10) for some constant C>0, then it can be checked that we have (1.3) with the same constant C. 2. It is possible to present a similar argument in a general frame. Let us consider three Hilbert spaces U,H,Eand two linear continuous operators L∈L(U;E) and M∈L(H;E). Then we have M∗ϕ0H≤CL∗ϕ0U∀ϕ0∈E (1.11) for some positive constant Cif and only if R(M)⊂R(L) and, moreover, ∀y0∈H, ∃v∈Usuch that Lv =My0,vU≤Cy0H.(1.12) Properties of this kind have been established and analyzed for the first time in the framework of control theory in [52]. They have been successfully used in many different contexts in recent years. 3. Because of linearity, the null controllability of (1.1) is equivalent to the exact controllability to (uncontrolled) trajectories. In other words, (1.1) is null controllable if and only if, for every y0∈L2(Ω), we can find a control v∈ L2(O×(0,T)) such that y(T)=y(T) in Ω, where yis the solution of (1.1) associated to v, and ysatisfies ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy=0 in Q, y= 0 on Σ, y(0) = y0in Ω. (1.13) Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1401 4. In general, for any y1∈L2(Ω), it is not reasonable to look for a control v∈L2(O×(0,T)) such that y(T)=y1in Ω (recall that O⊂⊂Ω). Indeed, due to the regularizing effect, every solution to (1.1) is space-analytic in Ω\Oat time T, which is not necessarily the case for y1. Thus, the exact controllability problem is out of order for the heat equation. In what follows, for the reasons stated above, we will focus on the proof of (1.3). This will rely on other (previous) inequalities for the solutions to (1.2), known as global Carleman inequalities. Let us mention that a general global Carleman inequality adopts the form Ω×(0,T ) ρ2|ϕ|2dx dt ≤CO×(0,T ) ρ2|ϕ|2dx dt,(1.14) where ρ=ρ(x, t) is a continuous and strictly positive weight function. For a function ρsatisfying ρ>0inΩ×(0,T), we will be able to deduce (1.14) and then estimates like Ω×(T/4,3T/4) |ϕ|2dx dt ≤CO×(0,T )|ϕ|2dx dt.(1.15) This, together with the dissipation properties of the solutions of (1.2), will lead to (1.3). More details are given below. Remark 2. To our knowledge, the first (boundary) controllability results for the heat equation were obtained by Egorov in the early sixties; see [17]. Later, an important general principle was established by Russell in [52]: if T0is sufficiently large and the wave system ⎧ ⎪ ⎨ ⎪ ⎩ ytt −Δy=v1Oin Ω ×(0,T 0), y=0 on ∂Ω×(0,T 0), y(0) = y0,y t(0) = y1in Ω (1.16) is exactly controllable at time T0, then (1.1) is null controllable at time Tfor all T>0. This principle has been revisited by several authors. In particular, an “abstract” version is given in [4]. Using the method of moments, the exact controllability of (1.16) can be established for large T0when, for instance, Ois a neighborhood of ∂Ω. Thus, one has null controllability for (1.1) in this case. Remark 3. In the particular case of the classical heat equation (1.1), there is another way to prove (1.3), based on the spectral decomposition of the solutions. This approach was introduced by Lebeau and Robbiano [41] and led to the null controllability of the heat equation with controls supported by general subdomains O⊂⊂Ω for the first time. 1.2. A global Carleman inequality for the linear heat equation and its consequences. Lemma 1.2. Let ω⊂⊂ Ωbe a nonempty open subset. Then there exists η0∈ C2(Ω) such that η0>0in Ω,η0=0on ∂Ω, and |∇η0|>0in Ω\ω. A proof of this lemma can be found in [30]. A much easier proof can be developed when Ω is star shaped with respect to a point x0∈ω. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1402 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO Let ωbe a nonempty open set satisfying ω⊂⊂ O and let us set α(x, t)=e2λmη0∞−eλ(mη0∞+η0(x)) t(T−t), ξ(x, t)=eλ(mη0∞+η0(x)) t(T−t) (1.17) for (x, t)∈Q, where η0is the function furnished by Lemma 1.2 for this ωand m>1. Weight functions of this kind were first introduced by Imanuvilov. See [30] for a systematic use of them. Now we arrive at the main result of this section. Lemma 1.3. There exist three constants λ1=C(Ω,O)≥1,s 1=C(Ω,O) (T+T2), and C1(Ω,O)such that, for any λ≥λ1and any s≥s1, the following inequality holds: s−1Q e−2sα ξ−1(|qt|2+|Δq|2)dx dt +sλ 2Q e−2sα ξ|∇q|2dx dt +s3λ4Q e−2sα ξ3|q|2dx dt ≤C1Q e−2sα |qt+Δq|2dx dt +s3λ4O×(0,T ) e−2sα ξ3|q|2dx dt (1.18) for all q∈C2(Q)with q=0on Σ. In what follows, C(Ω,O) or simply Cwill denote a generic constant depending only on Ω and O. Before giving the proof of Lemma 1.3, we will deduce the observability inequality (1.3) (and, accordingly, the null controllability of (1.1)) from this result. This can be made in three steps. Step 1. From the density of the smooth functions in the space where the solutions of (1.2) with ϕ0∈L2(Ω) live, we first observe that the Carleman inequality above is verified by all of them. Thus, fixing λ=λ1,weget Q e−2sα t−3(T−t)−3|ϕ|2dx dt ≤CO×(0,T ) e−2sα t−3(T−t)−3|ϕ|2dx dt (1.19) for all s≥s1. Step 2. Using the inequalities e−2s1αt−3(T−t)−3≥e−2C(Ω,O)(1+1/T )1 T6in Ω ×(T/4,3T/4)(1.20) and e−2s1αt−3(T−t)−3≤e−C(Ω,O)(1+1/T )1 T6in Ω ×(0,T),(1.21) we get Ω×(T/4,3T/4) |ϕ|2dx dt ≤C(Ω,O,T)O×(0,T )|ϕ|2dx dt,(1.22) Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1403 with a constant C(Ω,O,T) of the form eC(Ω,O)(1+1/T ). Step 3. From the equation verified by ϕ, we readily obtain ϕ(0)2 L2(Ω) ≤2 T3T/4 T/4ϕ(t)2 L2(Ω) dt.(1.23) This, together with (1.22), gives the observability inequality (1.3). In view of Theorem 1.1, we have the following. Theorem 1.4. The system (1.1) is null controllable with controls v∈L2(O× (0,T)) that verify vL2(O×(0,T )) ≤Cy0L2(Ω),(1.24) where the constant Cis of the form eC(Ω,O)(1+1/T ). Let us now give the proof of the Carleman estimate (1.18). The proof we present here is based on the ideas in [30]. We will try to keep an explicit dependence of all the constants with respect to the parameters sand λand the final time T. This follows the spirit of [27] and will be crucial for the analysis of similar nonlinear problems. Proof of Lemma 1.3. For simplicity, we will divide the proof into three steps. Step 1. Change of variables and plan of what follows. In this step, we set the differential equation satisfied by a new function ψ, which will be qup to a weight function. Thus, let us introduce the new functions ψ=e−sα qand g=e−sα f, where we have denoted f=qt+Δq. Then we easily obtain that M1ψ+M2ψ=gs,λ,(1.25) where M1ψ=−2sλ 2|∇η0|2ξψ−2sλξ∇η0·∇ψ+ψt, M2ψ=s2λ2|∇η0|2ξ2ψ+Δψ+sα tψ, (1.26) and gs,λ =g+sλΔη0ξψ−sλ 2|∇η0|2ξψ.(1.27) To simplify the notation, we will denote by (Miψ)j(1 ≤i≤2,1≤j≤3) the jth term in the expression of Miψgiven in (1.26). Observe that, at first sight, it seems “natural” to put (M1ψ)1on the right-hand side of (1.25). However, we have decided to keep it on the left, because it will serve to produce a positive term on |∇ψ|2(see the scalar product ((M1ψ)1,(M2ψ)2)L2(Q) below). With the previous notation, we have from (1.25) M1ψ2 L2(Q)+M2ψ2 L2(Q)+2 3  i,j=1 ((M1ψ)i,(M2ψ)j)L2(Q)=gs,λ2 L2(Q).(1.28) In the following steps, we will see that the definition we have made of αmakes 2(M1ψ,M2ψ)L2(Q)positive up to several terms that can be controlled whenever we make an appropriate choice of the parameters sand λ. More precisely, in the second step we will make the computations of the double products 2(M1ψ,M2ψ)L2(Q). This will give an inequality with two global terms of Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1410 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO the moment, we have s−1Q ξ−1|ψt|2dx dt +s−1Q ξ−1|Δψ|2dx dt +sλ 2Q ξ|∇ψ|2dx dt +s3λ4Q e−2sα ξ3|q|2dx dt ≤CQ e−2sα |f|2dx dt +s3λ4O×(0,T ) e−2sα ξ3|q|2dx dt. (1.64) Using that ∇q=esα(∇ψ−sλ∇η0ξψ),(1.65) we find sλ 2Q e−2sα ξ|∇q|2dx dt ≤Csλ 2Q ξ|∇ψ|2dx dt +Cs 3λ4Q e−2sα ξ3|q|2dx dt. (1.66) Consequently, we can add the previous integral of |∇q|2to the left-hand side of (1.64): s−1Q ξ−1|ψt|2dx dt +s−1Q ξ−1|Δψ|2dx dt +sλ 2Q ξ|∇q|2dx dt +s3λ4Q e−2sα ξ3|q|2dx dt ≤CQ e−2sα |f|2dx dt +s3λ4O×(0,T ) e−2sα ξ3|q|2dx dt. (1.67) For Δq, we use the identity Δψ=e−sα(Δq+sλΔη0ξq+sλ 2|∇η0|2ξq+2sλξ∇η0·∇q+s2λ2|∇η0|2ξ2q) (1.68) and obtain s−1Q e−2sα ξ−1|Δq|2dx dt ≤Cs−1Q ξ−1|Δψ|2dx dt +sλ 2Q e−2sα ξ|q|2dx dt +sλ 4Q e−2sα ξ|q|2dx dt +sλ 2Q e−2sα ξ|∇q|2dx dt +s3λ4Q e−2sα ξ3|q|2dx dt. (1.69) Finally, for qtwe get s−1Q e−2sα ξ−1|qt|2dx dt ≤Cs−1Q ξ−1|ψt|2dx dt +sT2Q e−2sα ξ3|q|2dx dt, (1.70) Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1411 where we have used the identity qt=esα(ψt+sα tψ).Thus, taking λ≥1 and s≥C(Ω,O)(T+T2), we are able to introduce all the terms involving |Δq|2and |qt|2on the left-hand side of (1.64). This gives (1.18) and concludes the proof of Lemma 1.3. Remark 4. It would be very interesting to know whether the powers of sand λarising in (1.18) are optimal. In other words, would it be possible to deduce an inequality like (1.18) with higher powers of sand λaccompanying the terms on the left-hand side? A positive answer to this would lead to the controllability of nonlinear parabolic systems more general than those considered in subsections 1.3.2 and 2.2.1. 1.3. Applications and generalizations. There are many other similar systems for which null controllability properties can be analyzed as before. Let us mention some of them. 1.3.1. General parabolic linear systems. We will now try to apply the Carleman inequality obtained in the previous subsection to systems where the heat equation has a zero order term and a first order term in the divergence form. More precisely, we would like to prove the null controllability of the system ⎧ ⎨ ⎩ yt−Δy+∇·(yB(x, t)) + a(x, t)y=v1Oin Q, y= 0 on Σ, y(0) = y0in Ω, (1.71) where y0∈L2(Ω),a∈L∞(Q), and B∈L∞(Q)N. In this case, the associated adjoint system is the following: ⎧ ⎨ ⎩ −ϕt−Δϕ−B(x, t)·∇ϕ+a(x, t)ϕ=0 in Q, ϕ= 0 on Σ, ϕ(T)=ϕ0in Ω. (1.72) A result similar to Theorem 1.1 holds for systems (1.71) and (1.72). Consequently, what we have to do is prove an observability inequality for the solutions to (1.72). In this situation, Lemma 1.3 provides a first estimate of the kind (1.18) (with q replaced by ϕ), with two additional terms on the right-hand side, namely, CQ e−2sα |aϕ|2dx dt and CQ e−2sα |B·∇ϕ|2dx dt.(1.73) But these can be absorbed by the left-hand side if we take slarge enough. Indeed, it suffices to take s≥C(Ω,O)T2(a2/3 ∞+B2 ∞)(1.74) to have CQ e−2sα |aϕ|2dx dt +Q e−2sα |B·∇ϕ|2dx dt ≤1 2s3λ4Q e−2sα ξ3|ϕ|2dx dt +1 2sλ 2Q e−2sα ξ|∇ϕ|2dx dt. (1.75) Arguing as we did when we proved the observability inequality for the solutions to (1.2), we easily find Ω×(T/4,3T/4) |ϕ|2dx dt ≤C(Ω,O,T,a,B)O×(0,T )|ϕ|2dx dt,(1.76) Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1412 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO with a constant C(Ω,O,T,a,B) = exp{C(1+1/T +a2/3 ∞+B2 ∞)}.(1.77) On the other hand, the dissipativity of ϕyields ϕ(0)2 L2(Ω) ≤exp{CT(a∞+B2 ∞)}ϕ(t)2 L2(Ω) (1.78) for all t∈(T/4,3T/4). Combining (1.76) and (1.78), we see that ϕ(0)2 L2(Ω) ≤eC(1+1/T +a2/3 ∞+Ta∞+(1+T)B2 ∞)O×(0,T )|ϕ|2dx dt(1.79) for all the solutions of (1.72) associated to final data ϕ0∈L2(Ω). As mentioned above (see Theorem 1.1), this implies the null controllability of (1.71). Theorem 1.5. System (1.71) is null controllable, with controls vsatisfying vL2(O×(0,T )) ≤C(Ω,O,T,a,B)y0L2(Ω),(1.80) where C(Ω,O,T,a,B)=eC(Ω,O)(1+1/T +a2/3 ∞+Ta∞+(1+T)B2 ∞).(1.81) Remark 5. In a recent paper, Seidman [54] looked at the qualitative asymptotic behavior of the constants found in estimates of the kind (1.80) with respect to relevant parameters. In particular, the blow-up of C(Ω,O,T,a,B)asT→0 has been analyzed in some particular cases. The next step will be to establish the null controllability for a general linear heat equation system with first order terms and coefficients in L∞(Q), i.e., an equation of the form yt−Δy+B(x, t)·∇y+a(x, t)y=v1O,(1.82) with a∈L∞(Q) and B∈L∞(Q)N. A different Carleman inequality is now required since, in the corresponding adjoint equation, there is the term −∇·(ϕB(x, t)), which belongs to L2(0,T;H−1(Ω)). This will be treated in section 2 using arguments from [39]. 1.3.2. Extension to some semilinear systems. We will now consider possible extensions of the null controllability results above to nonlinear problems of the form ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy+f(y)=v1Oin Q, y= 0 on Σ, y(0) = y0in Ω, (1.83) where f∈C1(R) (for instance),f(0)=0,and |f(s)|≤C(1 + |s|).(1.84) The classical strategy used in [36] and [30] relies on the introduction of a mapping z∈L2(Q)→ yz∈L2(Q),(1.85) Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1413 where yzis, together with vz, a solution of ⎧ ⎨ ⎩ (yz)t−Δyz+g(z)yz=vz1Oin Q, yz= 0 on Σ, yz(0) = y0in Ω (1.86) such that yz(T)=0(1.87) and (1.80) holds for vzwith a=g(z) and B≡0. Here, gis given by g(s)=⎧ ⎨ ⎩ f(s) sif s=0, f(0) if s=0. (1.88) Using the previous results and arguing as in [36], it can be seen that this mapping can be correctly defined. It can also be proved that it is continuous and compact. Furthermore, in view of (1.84), the estimate (1.80) written for vz, and the classical estimates for yz, it maps the whole space L2(Q) into a ball. Hence, from Schauder’s theorem, we deduce that this mapping possesses a fixed point and, consequently, the following holds. Theorem 1.6. Under conditions (1.84) on f, system (1.83) is null controllable, with controls vsatisfying vL2(O×(0,T )) ≤exp{C(Ω,O)(1+1/T +g2/3 ∞+Tg∞)}y0L2(Ω),(1.89) where gis given by (1.88). This fixed point approach for the solution of controllability problems for nonlinear systems was introduced in [56] in the context of the controllability of the wave equation. Later, it was successfully applied to the semilinear heat equation in [36], [19], [30], [20], and [28]. The next case corresponds to a superlinear f, i.e., to a function not satisfying the last condition in (1.84). In this context, it is much more complicated to estimate the norms of vzand yz, since gis not necessarily uniformly bounded. However, something can still be made in some particular cases. More precisely, the following result is proved in [28]. Theorem 1.7. Assume that ⎧ ⎪ ⎨ ⎪ ⎩ f∈C1(R)(for instance),f(0)=0,and |f(s)| slog3/2(1 + |s|)−→ 0as |s|→+∞. (1.90) Then (1.83) is null controllable with controls v∈L∞(O×(0,T)). Remark 6. It is also proved in [28] that, in general, when |f(s)|∼|s|logβ(1 + |s|)(1.91) for some β>2, (1.83) is not null controllable. However, it is unknown whether or not the system (1.83) is null controllable when this is satisfied with 3 2<β≤2. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1414 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO 1.3.3. Another boundary conditions. Let us consider the following system, where the heat equation is completed with linear Robin (or Fourier) boundary conditions: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ yt−Δy=v1Oin Q, ∂y ∂n +a(x, t)y= 0 on Σ, y(0) = y0in Ω, (1.92) where a∈L∞(Σ) and y0∈L2(Ω). Recall that under these assumptions there exists a unique solution yto (1.92), with y∈L2(0,T;H1(Ω)) ∩C0([0,T]; L2(Ω)). Arguing as above, it is readily seen that the null controllability of (1.92) is implied by the observability of the adjoint system ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −ϕt−Δϕ=0 in Q, ∂ϕ ∂n +a(x, t)ϕ= 0 on Σ, ϕ(T)=ϕ0in Ω. (1.93) An observability inequality for the solutions to (1.93) is proved in [30], under the additional assumption at∈L∞(Σ). As a consequence, one also has the null controllability of (1.92) whenever afulfills this hypothesis. For some partial results concerning a nonlinear version of (1.92), see [15]. An improvement of these results has been obtained in the more recent work [21]. See section 2 for further details. 1.3.4. Another controllability problems. Another interesting problem is the null controllability of the heat equation with boundary controls, which means the null controllability of the system ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy=0 in Q, y=v1γon Σ, y(0) = y0in Ω, (1.94) when γis a (small) part of ∂Ω. In this case, the adjoint system is again (1.2) but the required observability inequality is somewhat different. More precisely, we need the estimate ϕ(0)2 L2(Ω) ≤Cγ×(0,T ) ∂ϕ ∂n 2 dσ dt(1.95) for all solutions to (1.2). This can be obtained as a consequence of the following Carleman estimates, similar to (1.18) (see [30]): Q e−2sα(s ξ)−1(|qt|2+|Δq|2)+sλ 2 ξ|∇q|2+s3λ4 ξ3|q|2dx dt ≤CQ e−2sα|qt+Δq|2dx dt +sγ×(0,T ) e−2sα ξ ∂q ∂n 2 dσ dt. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1415 Here, α=α(x, t) and  ξ= ξ(x, t) are appropriate weight functions, similar to αand ξ, respectively. Using (1.95) and arguments like those in the proof of Theorem 1.1, one can easily deduce the null controllability of (1.94). 1.3.5. Other more general partial differential equations. The previous null controllability results can also be extended to more general equations with sufficiently regular coefficients. For instance, for the system ⎧ ⎪ ⎨ ⎪ ⎩ yt−∂i(aij(x, t)∂jy)=v1Oin Q, y= 0 on Σ, y(0) = y0in Ω, (1.96) where the coefficients aij ∈C2(Q) are uniformly elliptic, it is proved in [30] that null controllability is achieved. This result has recently been improved in [51]; see subsection 2.2.3 below. 2. Null controllability of other more general parabolic problems. 2.1. The case of the linear heat equation with general zero and first order terms. In this subsection, we are going to prove the null controllability, with distributed controls, of the linear heat equation with zero and first order terms and coefficients in L∞(Q), i.e., ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy+B(x, t)·∇y+a(x, t)y=v1Oin Q, y= 0 on Σ, y(0) = y0in Ω, (2.1) with y0∈L2(Ω),a∈L∞(Q), and B∈L∞(Q)N. To achieve this, we will first prove an observability inequality for the associated adjoint system ⎧ ⎪ ⎨ ⎪ ⎩ −ϕt−Δϕ−∇·(ϕB(x, t)) + a(x, t)ϕ=0 in Q, ϕ= 0 on Σ, ϕ(T)=ϕ0in Ω. (2.2) More precisely, we will prove that ϕ(0)2 L2(Ω) ≤CO×(0,T )|ϕ|2dx dt(2.3) for all ϕ0∈L2(Ω) for some C=C(Ω,O,T,a,B). To this end we will use the following lemma, which provides a suitable Carleman inequality (see [39]). Lemma 2.1. There exist λ2=C(Ω,O),s 2=C(Ω,O)(T+T2), and C2= C2(Ω,O)>0such that, for any λ≥λ2and any s≥s2, one has sλ 2Q e−2sα ξ|∇q|2dx dt +s3λ4Q e−2sα ξ3|q|2dx dt ≤C2s3λ4O×(0,T ) e−2sα ξ3|q|2dx dt +Q e−2sα |F0|2dx dt +s2λ2Q e−2sα ξ2|F|2dx dt (2.4) Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1416 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO for all functions q∈C1(Q)with q=0on Σand qt+Δq=F0+∇·F, where F0∈ L2(Q),F∈L2(Q)N. Here, αand ξare the functions defined just before Lemma 1.3. For the moment, let us assume that Lemma 2.1 is true and let us deduce an observability inequality of the kind (2.3) for the solutions to (2.2). Fixing λ=λ2and applying Lemma 2.1 to ϕ,wehave s3Q e−2sα ξ3|ϕ|2dx dt +sQ e−2sα ξ|∇ϕ|2dx dt ≤Cs3O×(0,T ) e−2sα ξ3|ϕ|2dx dt +a2 ∞Q e−2sα |ϕ|2dx dt +s2B2 ∞Q e−2sα ξ2|ϕ|2dx dt (2.5) for any s≥s2. Consequently, we also have Q e−2sα t−3(T−t)−3|ϕ|2dx dt ≤CO×(0,T ) e−2sα t−3(T−t)−3|ϕ|2dx dt(2.6) for all s≥C(T+T2+T2(a2/3 ∞+B2 ∞)), whence it is easy to deduce that Ω×(T/4,3T/4) |ϕ|2dx dt ≤eC(1+1/T +a2/3 ∞+B2 ∞)O×(0,T )|ϕ|2dx dt.(2.7) On the other hand, we also have (1.78) for the solutions to (2.2). This, together with (2.7), leads to the desired observability inequality (2.3). Thus, we have the following. Theorem 2.2. System (2.1) is null controllable with controls vsatisfying (1.80) and (1.81). Following [39], let us now give the proof of the Carleman inequality (2.4). Proof of Lemma 2.1.Let q=q(x, t) satisfy the hypothesis stated above and set q0=q|t=T. Then qcan be viewed as a solution by transposition of the backwards system ⎧ ⎨ ⎩ qt+Δq=F0+∇·Fin Q, q= 0 on Σ, q(T)=q0in Ω. (2.8) In other words, for each G∈L2(0,T;H−1(Ω)) we must have T 0G(t),q(t)dt =−T 0F0(t)+∇·F(t),z(t)dt +(q0,z(T))L2(Ω),(2.9) where zis the solution of the linear problem ⎧ ⎪ ⎨ ⎪ ⎩ zt−Δz=Gin Q, z= 0 on Σ, z(0) = 0 in Ω. (2.10) Here, ·,· denotes the usual duality product between H−1(Ω) and H1 0(Ω). Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1417 Let sand λbe as in Lemma 1.3 and let us introduce the following fourth order problem, which will be justified below: ⎧ ⎪ ⎨ ⎪ ⎩ L(e−2sαL∗p)+s3λ4e−2sα ξ3q=−s3λ4e−2sα ξ3p1Oin Q, p=0,e −2sαL∗p= 0 on Σ, (e−2sαL∗p)(0) = (e−2sαL∗p)(T)=0 in Ω. (2.11) Here, we have used the notation Lq≡qt−Δq, L∗q≡−qt−Δq. The partial differential problem (2.11) possesses exactly one (weak) solution pwith Q e−2sα(ξ−1(|pt|2+|Δp|2)+ξ|∇p|2+ξ3|p|2)dx dt < +∞.(2.12) Indeed, let P0be the linear space P0={z∈C2(Q):z= 0 on Σ}(2.13) and let us set ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ κ(p, p)=Q e−2sα L∗pL∗pdx dt +s3λ4O×(0,T ) e−2sα ξ3pp dx dt ∀p, p∈P0 (2.14) and l(p)=−s3λ4Q e−2sα ξ3q p dx dt ∀p∈P0.(2.15) Then κ(·,·) is a positive and symmetric bilinear form in P0. Let Pbe the completion of P0for the norm pP=(κ(p, p))1/2.Then Pis a Hilbert space for the scalar product κ(·,·) and, in view of the Carleman inequality (1.18), we have that the functions in Psatisfy (2.12). It is also clear from (1.18) that lis a continuous linear form on P: |l(p)|≤Cs3λ4Q e−2sα ξ3|q|2dx dt1/2 pP∀p∈P.(2.16) Consequently, from the Lax–Milgram lemma, the following variational equation possesses exactly one solution p∈P: κ(p, p)=l(p)∀p∈P.(2.17) It is not difficult to see that the unique solution to (2.17) also solves the fourth order problem (2.11) in the distributional sense. Of course, the space Pand the function p∈Pdepend on the choice we have made of sand λ. Now, let us set z=−e−2sα L∗p, u=s3λ4e−2sα ξ3p1O.(2.18) Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1418 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO It is readily seen from (2.11) that zis, together with u, a solution to the null controllability problem ⎧ ⎪ ⎨ ⎪ ⎩zt−Δz=s3λ4e−2sα ξ3q+u1Oin Q, z= 0 on Σ, z(0) = z(T)=0 in Ω. (2.19) For the moment, we will assume that there exist positive s=s(Ω,O) and  λ= λ(Ω,O), such that s−3λ−4O×(0,T ) e2sα ξ−3|ˆu|2dx dt +Q e2sα |ˆz|2dx dt +s−2λ−2Q e2sα ξ−2|∇ˆz|2dx dt ≤Cs 3λ4Q e−2sα ξ3|q|2dx dt (2.20) for all s≥s(T+T2) and λ≥ λ. Let us then prove (2.4). First, we will get an estimate for the second term on the left-hand side of (2.4). In view of (2.9) written for G=s3λ4e−2sα ξ3q+u1O,we have s3λ4Q e−2sα ξ3|q|2dx dt =−O×(0,T ) qudxdt+T 0F0(t)+∇·F(t),z(t)dt. (2.21) Therefore, s3λ4Q e−2sα ξ3|q|2dx dt ≤T 0O e−2sα ξ3|q|2dx dt1/2T 0O e2sα ξ−3|u|2dx dt1/2 +Q e−2sα|F0|2dx dt1/2Q e2sα|z|2dx dt1/2 +Q e−2sα ξ2|F|2dx dt1/2Q e2sα ξ−2|∇z|2dx dt1/2 . (2.22) And now, using (2.20), we see that s3λ4Q e−2sα ξ3|q|2dx dt ≤Cs3λ4O×(0,T ) e−2sα ξ3|q|2dx dt +Q e−2sα |F0|2dx dt +s2λ2Q e−2sα ξ2|F|2dx dt (2.23) for s≥s(T+T2) and λ≥ λ. Let us now get an estimate for the first term on the left-hand side of (2.4). To this end, we multiply by sλ 2e−2sα ξq the equation satisfied by qand integrate in space Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1419 and time. This gives sλ 2Q e−2sα ξ|∇q|2dx dt +sλ2 2Q(e−2sα ξ)t−Δ(e−2sα ξ)|q|2dx dt =sλ 2Q e−2sα(ξF 0q−ξF·∇q)dx dt −sλ 2Q F·∇(e−2sα ξ)q dx dt. (2.24) Then we use the estimates |(e−2sα ξ)t|≤Ce −2sα (sTξ3+Tξ 2)≤Cs 2e−2sα ξ3, |Δ(e−2sα ξ)|≤Cs 2λ2e−2sα ξ3 (2.25) for s≥C(T+T2), sλ 2Q e−2sα ξF 0qdxdt ≤Cs3λ4Q e−2sαξ3|q|2dx dt +s−1Q e−2sαξ−1|F0|2dx dt, sλ 2Q e−2sα ξF·∇qdxdt ≤Csλ 2Q e−2sα ξ|F|2dx dt +1 2sλ 2Q e−2sα ξ|∇q|2dx dt, (2.26) and sλ 2Q F·∇(e−2sα ξ)qdxdt  ≤Csλ 2Q e−2sα ξ|F|2dx dt +s3λ4Q e−2sα ξ3|q|2dx dt, (2.27) to obtain from (2.24) the following: sλ 2Q e−2sα ξ|∇q|2dx dt ≤Cs3λ4Q e−2sα ξ3|q|2dx dt +Q e−2sα |F0|2dx dt +sλ 2Q e−2sα ξ|F|2dx dt (2.28) for s≥C(T+T2). This, together with (2.23), provides (2.4). Let us finally prove (2.20). We first multiply the equation in (2.11) by p,sowehave χ(p, p)=−s3λ4Q e−2sα ξ3q p dx dt,(2.29) which, combined with the Carleman inequality (1.18) proved in Lemma 1.3, provides the desired inequality for the two first terms on the left-hand side of (2.20). Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1426 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO whose adjoint system is the following: ⎧ ⎪ ⎨ ⎪ ⎩ −(bϕ)t−Δϕ=0 in Q, ϕ= 0 on Σ, ϕ(T)=ϕ0in Ω. (2.64) In order to analyze the null controllability of (2.63), we need a Carleman inequality for heat equations with right-hand sides in H−1(0,T;L2(Ω)). Its proof can also be found in [23]. Lemma 2.12. Consider the system ⎧ ⎪ ⎨ ⎪ ⎩ −zt−Δz=GtQ, z=0Σ, z(T)=z0Ω, (2.65) where z0∈H−1(Ω) and G∈L2(Q)with G∈Cw([0,T]; H−1(Ω)). Then there exist C4(Ω,O)>0,λ 4=C(Ω,O), and s4=C(Ω,O)(T+T2)such that s3λ4Q e−2sα ξ3|z|2dx dt ≤C4s3λ4O×(0,T ) e−2sα ξ3|z|2dx dt +s4λ4Q e−2sα ξ4|G|2dx dt (2.66) for all λ≥λ4and s≥s4. From this Carleman inequality, one can readily deduce the following result. Theorem 2.13. Let Tbe a positive time and let us assume that b=1+εh with (T−t)−1/2h∈L∞(Q)and ε>0small enough, depending on Ω,O, and T. Then system (2.63) is null controllable at time T. Remark 9. The proofs of Lemmas 2.10 and 2.12 are based on the duality arguments presented in Lemma 2.1. Observe in particular that the solutions to systems (2.58) and (2.63) belong only to L2(Q) and must be defined by transposition. 3. Null controllability of systems of the Stokes kind. In this section, N= 2orN= 3. The controllability properties of the Navier–Stokes system have been the subject of intensive research these last years. The question was first considered by Lions in [48], where approximate controllability was conjectured. This was followed by several papers, where various partial (positive) answers were furnished. See [18], [10], [11], and [50]. Concerning null controllability and exact controllability to the trajectories, the first local results were given in [29]. For completeness, we will recall briefly one of the main results in [10]. This paper deals with the approximate controllability of the two-dimensional Navier–Stokes equations with Navier-slip boundary conditions on the lateral boundary ∂Ω×(0,T). Thus, let us assume that the state equation is ⎧ ⎪ ⎨ ⎪ ⎩ yt+(y·∇)y−Δy+∇p=v1O,∇·y=0 in Q, y·n=0,∇×y= 0 on Σ, y(0) = y0in Ω (3.1) (where Ω ⊂R2). The general fact the author makes good use of is the treatment of (y·∇)yas the leading term, while −Δyis considered as a “disturbance.” Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1427 Roughly speaking, the following is proved: Let y0and y1be two prescribed velocity fields. Then the fluid can be driven from y0at time t= 0 to a velocity field at time t=Twhich is arbitrarily close to y1. Here, “arbitrarily close” means that, for instance, the distance of the velocity field to y1in W−1,∞(Ω) and the distance of the vorticity to ∇×y1in all spaces L∞(K)(K⊂Ω is an arbitrary compact set) can be made arbitrarily small. This is proved for any T>0, for any regular Ω, y0, and y1, and for all nonempty open sets O⊂Ω. The proof is as follows: 1. For each α>0, it is possible to find vαand a couple (yα,p α) such that yαis a gradient outside O×(0,T), ⎧ ⎪ ⎨ ⎪ ⎩ yα,t +(yα·∇)yα−Δyα+∇pα=vα1O,∇·yα=0 in Q, yα·n=0,∇×yα= 0 on Σ, yα(0) = yα(T)=0 in Ω (3.2) and, furthermore, the linearized Euler system at yα, that is to say, zt+(yα·∇)z+(z·∇)yα+∇π=w1O,∇·z=0 in Q, z·n=0,∇×z= 0 on Σ, (3.3) is α-controllable in the following sense: For any given z0and zdof class C∞, there exists a control wsuch that (3.3) possesses at least one solution bounded in C3(Q) independently of αand satisfying z(0) = z0in Ω and z(T)=zdin {x∈Ω; dist (x, ∂Ω) ≥α} (notice we are saying that, for every α, this property is satisfied for any z0and zd). 2. Let us set v(x, t) = 0 in (3.1) for t∈[0,(1 −δ)T](δwill be determined below). This defines yand pwithout ambiguity in Ω ×[0,(1 −δ)T] and, in particular, we can speak of y((1 −δ)T). In [(1 −δ)T,T], we do the following: •First, vα,yα, and pαare rescaled. In view of the α-controllability of (3.3), we introduce a first control function v: v(x, t)≡1 δvαx, 1 δ(t−(1 −δ)T)+wx, 1 δ(t−(1 −δ)T). The associate state is (y, p)≡1 δ(yα,p α)x, 1 δ(t−(1 −δ)T)+(z,π)x, 1 δ(t−(1 −δ)T). Here, wand (z,π) are perturbations depending on α(which will be also fixed below), the initial state y((1 −δ)T), and the desired state yd. In order to drive (3.1) to a final state close to yd, it is natural (at least formally) to look for a control close to vfor t∈[(1 −δ)T,T]. •We introduce a second control function by modifying vjust as needed. Thus, we solve the following problem: ⎧ ⎪ ⎨ ⎪ ⎩ yt+(y·∇)y−Δy+∇p=v1O+(∇×y)(y−y)⊥,∇·y=0 in Q, y·n=0,∇×y= 0 on Σ, y(0) = y0in Ω. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1428 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO Here, we have introduced the function y(x, t)=1 δyαx, 1 δ(t−(1 −δ)T),(a1,a 2)⊥=(−a2,a 1). Since ∇×yvanishes outside O×(0,T), it is clear that (y,p) solves (3.1) with v=v+(∇×y)(y−y)⊥. Now, the task is reduced to show that for every ε>0, there exist positive αand δsuch that y(T)−ydW−1,∞≤ε.(3.4) This can be achieved in the following way. Let us set R=y−y0,ω=∇×R. Then ∂tω+(R+y0)·∇ω−Δω=−(R+z)·∇(∇×z)+Δ(∇×z) in Ω ×((1 −δ)T,T). Furthermore, R(x, (1 −δ)T)≡0 and zand all its derivatives of order ≤3 are uniformly bounded. This leads, at first time, to a pointwise estimate of ωand then to an estimate of R(·,T)inW−1,∞(Ω) when αand δare sufficiently small. Consequently, for any given ε>0, there exist α0>0 and η:(0,α 0)→ R+ such that whenever 0 <α<α 0and 0 <δ<η(α), one has (3.4) (for further details, see [10]). This method has several limitations. Thus, the boundary conditions have to be of the Navier-slip type. In practice, this is equivalent to prescribing the values on the boundary of the stream function and the vorticity. At present, it is unknown how the method has to be modified in order to keep its validity in the context of Dirichlet (no-slip) boundary conditions. The case in which Ω is a manifold without boundary is considered in [11]. An extension of this method to Boussinesq systems in spatial dimensions N=2 and N= 3 has been given in [31]. For the analysis of the similar three-dimensional situation for the Euler equations, see [33]. In the remainder of this section, we will be concerned with the null controllability of Stokes-like and Navier–Stokes systems completed with Dirichlet boundary conditions. Let us recall the definition of the following spaces, which are usual in the analysis of Stokes systems: V={z∈C∞ 0(Ω)N:∇·z= 0 in Ω } and H(resp., V) = the adherence of Vin L2(Ω)N(resp., H1 0(Ω)N). We have H={z∈L2(Ω)N:∇·z= 0 in Ω,z·n=0on∂Ω}(3.5) and V={z∈H1 0(Ω)N:∇·z= 0 in Ω }.(3.6) For instance, see [55]. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1429 3.1. Null controllability of Stokes systems. We are first interested in proving the null controllability of some systems of the Stokes kind with distributed control. In other words, we would like to be able to find v,y, and psuch that ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ yt−Δy+∇·(y⊗y)+∇·(y⊗y)+∇p=v1Oin Q, ∇·y=0 in Q, y= 0 on Σ, y(0) = y0in Ω (3.7) and y(T)=0 inΩ(3.8) for any y0∈H. Here, we will assume that the following regularity properties hold: y∈L∞(Q)N, yt∈L2(0,T;Lσ(Ω))N,(3.9) with σ>1ifN= 2 and σ>6/5ifN=3. Let us remark that (3.7) can be viewed as the linearized Navier–Stokes system, so we are actually trying to prove the exact controllability to the trajectories of those equations. In fact, this will serve to deduce the local exact controllability to the trajectories of the Navier–Stokes equations. A previous result of this kind was proved in [37] under stronger hypotheses on the trajectory y. In what follows, we will sketch the arguments presented in [24], which are also based on those in [37]. Concerning the null controllability of (3.7), we will first recall a Carleman inequality for the backwards system ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −ϕt−Δϕ−(Dϕ)y+∇π=0 in Q, ∇·ϕ=0 in Q, ϕ= 0 on Σ, ϕ(T)=ϕ0in Ω, (3.10) where Dϕ =∇ϕ+∇ϕtand, consequently, the ith component of (Dϕ)yis ((Dϕ)y)i= N  j=1 (∂jϕi+∂iϕj)yj.(3.11) To this end, let us introduce the following weight functions: β(x, t)=e5/4λkη∞−eλ(kη∞+η(x)) t4(T−t)4,γ(x, t)=eλ(kη∞+η(x)) t4(T−t)4,  β(t) = min x∈Ω β(x, t),β ∗(t) = max x∈Ω β(x, t), γ(t) = max x∈Ω γ(x, t),γ ∗(t) = min x∈Ω γ(x, t). (3.12) Here, k>4 is a fixed number and ηis the function given by Lemma 1.2 associated to ω=O5⊂⊂ O. Then we have the following. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1430 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO Lemma 3.1. Let us assume that (3.9) is satisfied. Then for every ϕ0∈Hthere exist three positive constants s, λ, and Cdepending only on Ωand O, such that s−1Q e−2sβ γ−1|ϕt|2dx dt +s−1Q e−2sβ γ−1|Δϕ|2dx dt +sλ2Q e−2sβ γ|∇ϕ|2dx dt +s3λ4Q e−2sβ γ3|ϕ|2dx dt ≤Cs 16λ40(1 + T2)O×(0,T ) e−8s β+6sβ∗γ16 |ϕ|2dx dt (3.13) for all λ≥λ(1 + y∞+yt2 L2(Lσ)+eλT y2 ∞)and s≥s(T4+T8). The proof of Lemma 3.1 is given in the next subsection. Notice that the hypotheses imposed above over yare not completely satisfactory. In fact, it is expected (and desirable) that an inequality like (3.13) hold for any y∈L∞(Q)N, but for the moment this is unknown. Now, from this Carleman inequality, one can deduce the null controllability of (3.7) in a classical way. Theorem 3.2. Let us assume that (3.9) holds. Then for any T>0, the linear system (3.7) is null controllable with controls v∈L2(O×(0,T))Nat time T. After some additional work, one can also deduce some local controllability results for the Navier–Stokes equations. Specifically, we mean local exact controllability to the trajectories. Thus, let us introduce the Navier–Stokes system ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ yt−Δy+∇·(y⊗y)+∇p=v1Oin Q, ∇·y=0 in Q, y= 0 on Σ, y(0) = y0in Ω. (3.14) We have the following. Theorem 3.3. Let (y,p)be a solution to the Navier–Stokes problem ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ yt−Δy+∇·(y⊗y)+∇p=0 in Q, ∇·y=0 in Q, y=0 on Σ, y(0) = y0in Ω (3.15) verifying (3.9), and let us introduce the space E, with E=Hif N=2 and E=H∩L4(Ω)3if N=3.(3.16) Then there exists ε>0such that for each y0∈Esatisfying y0−y0E≤ε, there exist controls v∈L2(O×(0,T))Nand associated states (y,p)such that one has (3.14) and y(T)=y(T)in Ω.(3.17) This result can be proved by arguing as follows. We deduce in a first step a null controllability result for (3.7) with suitable right-hand side f. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1431 More precisely, let us set Ly =yt−Δy+∇·(y⊗y+y⊗y) and let us introduce the spaces EN, with E2={(y,v):e2sb−sb∗γ−15/4y, e4sb−3sb∗γ−8v1ω∈L2(Q)2, esb∗/2(γ∗)−1/4y∈L2(0,T;V)∩L∞(0,T;H), ∃p:esb∗(γ∗)−1/2(Ly +∇p−v1ω)∈L2(0,T;H−1(Ω)2)} and E3={(y,v):e2sb−sb∗γ−15/4y, e4sb−3sb∗γ−8v1ω∈L2(Q)3, esb∗/2(γ∗)−1/4y∈L2(0,T;V)∩L∞(0,T;H), esb∗/2(γ∗)−1/4y∈L4(0,T;L12(Ω)3), ∃p:esb∗(γ∗)−1/2(Ly +∇p−v1ω)∈L2(0,T;W−1,6(Ω)3)}, where the new weight functions b,b∗, etc., are given by b(x, t)=e5/4λmη0∞−eλ(mη0∞+η0(x)) (t)4, b(t) = min x∈Ω b(x, t),b ∗(t) = max x∈Ω b(x, t), γ(x, t)=eλ(mη0∞+η0(x)) (t)4, γ(t) = max x∈Ω γ(x, t),γ ∗(t) = min x∈Ω γ(x, t). Here, we have introduced (t)=T2/4 for 0 <t<T/2, t(T−t) for T/2<t<T. We then have the following. Proposition 3.4. Assume that ysatisfies (3.9) and •y0∈H,esb∗(γ∗)−1/2f∈L2(0,T;H−1(Ω)2)if N=2; •y0∈L4(Ω)3∩H,esb∗(γ∗)−1/2f∈L2(0,T;W−1,6(Ω)3)if N=3. Then there exists a control v∈L2(ω×(0,T))Nsuch that if yis the associated solution to (3.7), we have (y,v)∈EN. Notice that this is actually a null controllability result for (3.7). Indeed, if (y,v)∈ EN, we have in particular that y(T) = 0 in Ω. The rest of the proof of Theorem 3.3 relies on an appropriate inverse mapping theorem. More precisely, we use the following result from [1]. Proposition 3.5. Let X, F be two Banach spaces and let A:X→ Fsatisfy A∈ C1(X;F). Assume that e0∈X,A(e0)=h0, and A(e0):X→ Fis an epimorphism. Then there exists δ>0such that for every h∈Fsatisfying h−h0F<δ, there exists a solution of the equation A(e)=h, e ∈X. Let us consider the mapping A:X→ F, given by A(y,v)=(Ly +(y·∇)y+∇p−v1ω,y(·,0)) ∀(y,v)∈X, Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1432 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO where X=ENand F=L2(esb∗(γ∗)−1/2;H−1(Ω)2)×Hif N=2, L2(esb∗(γ∗)−1/2;W−1,6(Ω)3)×(L4(Ω)3∩H)ifN=3. From the definition of EN, one can easily check that Ais well defined and satisfies A∈C1(X;F). Furthermore, the identity Im(A(0,0)) = F is equivalent to the result stated in Proposition 3.4. Therefore, we can apply Proposition 3.5 to Awith e0=(0,0) and h0=(0,0). This ends the proof of Theorem 3.3. For more details concerning the local exact controllability of (3.14), see [24]; see also [25]. 3.2. Proof of the Carleman inequality. In this subsection, we sketch the proof of Lemma 3.1. This is contained in a more general case covered by the results in [24]. Throughout the proof, Cwill denote a generic positive constant that may depend on Ω and Oand can change its value from one line to the next. For easier comprehension, we will divide the proof in several steps. 3.2.1. Carleman estimate for the heat equation and first estimate of the pressure. Let us apply the Carleman inequality (1.18) to each component ϕi with right-hand side −∂iπ+((Dϕ)y)i(1 ≤i≤N).(3.18) Introducing the notation I(s, λ;ϕ)=s−1Q e−2sb γ−1(|ϕt|2+|Δϕ|2)dx dt +sλ2Q e−2sβ γ|∇ϕ|2dx dt +s3λ4Q e−2sβ γ3|ϕ|2dx dt, (3.19) one has I(s, λ;ϕ)≤Cs3λ4O5×(0,T ) e−2sβ γ3|ϕ|2dx dt +Q e−2sβ (|∇π|2+|(Dϕ)y|2)dx dt (3.20) for λ≥Cand s≥C(T7+T8). Taking λ≥Cy∞and s≥CT8, one can eliminate the term |(Dϕ)y|2on the right-hand side, whence we obtain I(s, λ;ϕ)≤Cs3λ4O5×(0,T ) e−2sβ γ3|ϕ|2dx dt +Q e−2sβ |∇π|2dx dt (3.21) for λ≥C(1 + y∞) and s≥C(T4+T8). Let us now present an estimate of the last integral. To this end, we will apply the main result in [38] to the elliptic equation −Δπ(t)=−∇·((Dϕ)y)(t)inΩ(3.22) Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1433 for each t∈(0,T). This provides Ω e2τeλη |∇π(t)|2dx ≤CO5 e2τeλη (|∇π(t)|2+τ2λ2e2λη |π(t)|2)dx +τΩ e2τeλη eλη |(Dϕ)y(t)|2dx +τ1/2e2τπ(t)2 H1/2(∂Ω) (3.23) for λ, τ ≥C. One can readily see that the local term of |∇π|2can be eliminated by arguing as in the proof of Lemma 1.3, so we obtain Ω e2τeλη |∇π(t)|2dx ≤Cτ2λ2O4 e2τeλη e2λη |π(t)|2dx +τΩ e2τeλη eλη |(Dϕ)y(t)|2dx +τ1/2e2τπ(t)2 H1/2(∂Ω) (3.24) for an open subset O4verifying O5⊂⊂ O4⊂⊂ O. Next, we put τ=s t4(T−t)4eλkη∞,(3.25) we multiply (3.24) by exp −2se5/4λkη∞ t4(T−t)4,(3.26) and we integrate with respect to tin (0,T). This yields Q e−2sβ |∇π|2dx dt ≤Cs2λ2O4×(0,T ) e−2sβ γ2|π|2dx dt +sQ e−2sβ γ|(Dϕ)y|2dx dt +s1/2T 0 e−2sβ∗(γ∗)1/2π(t)2 H1/2(∂Ω) dt (3.27) for λ≥Cand s≥CT8. Let us now estimate the term containing the trace of πfollowing the ideas of [37]. Thus, let us introduce the functions ϕ∗=s1/4e−sβ∗(γ∗)1/4ϕand π∗=s1/4e−sβ∗(γ∗)1/4π,(3.28) which satisfy ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −ϕ∗ t−Δϕ∗+∇π∗=(Dϕ∗)y−s1/4(e−sβ∗(γ∗)1/4)tϕin Q, ∇·ϕ∗=0 in Q, ϕ∗= 0 on Σ, ϕ∗(T)=0 in Ω. (3.29) Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1434 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO Using well-known regularity properties of the Stokes system, we find π∗2 L2(0,T ;H1/2(∂Ω)) ≤CQ (|Dϕ∗y|2+s1/2|(e−sβ∗(γ∗)1/4)t|2|ϕ|2)dx dt ≤Cs1/2Q e−2sβ∗(γ∗)1/2|(Dϕ)y|2dx dt +s5/2T2Q e−2sβ∗(γ∗)3|ϕ|2dx dt (3.30) for s≥CT8. This, used in combination with (3.27), leads to the inequality Q e−2sβ |∇π|2dx dt ≤Cs2λ2O4×(0,T ) e−2sβ γ2|π|2dx dt +sQ e−2sβ γ|(Dϕ)y|2dx dt +s5/2T2Q e−2sβ γ3|ϕ|2dx dt, (3.31) which must hold for λ≥Cand s≥CT8. We finally insert this inequality into (3.21) and absorb the last two terms on the right-hand side: I(s, λ;ϕ)≤Cs 2λ2O4×(0,T ) e−2sβ(sλ2γ3|ϕ|2+γ2|π|2)dx dt(3.32) for λ≥C(1 + y∞) and s≥C(T4+T8). 3.2.2. Local estimate of the pressure. The goal of this paragraph is to estimate the local integral of πappearing in (3.32). This is the most complicated part of the proof. Let us assume that we have chosen πsuch that O4 π(t)dx =0 ∀t∈(0,T).(3.33) Then from the definition of the weight functions and the Poincar´e–Wirtinger inequality, we have s2λ2O4×(0,T ) e−2sβ γ2|π|2dx dt ≤s2λ2O4×(0,T ) e−2s βγ2|π|2dx dt ≤Cs 2λ2O4×(0,T ) e−2s βγ2|∇π|2dx dt. (3.34) Recall that  β= β(t) and γ=γ(t) are given as follows:  β(t) = min Ω β(x, t),γ(t) = max Ω γ(x, t).(3.35) From the differential equation in (3.10), we deduce that in order to estimate the last integral in (3.34), it suffices to bound locally the Laplacian and the time derivative of the velocity field ϕ. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1435 A local estimate of Δϕ.Let us first introduce two open sets O2and O3, with O4⊂⊂ O3⊂⊂ O2⊂⊂ O,(3.36) and a function ρ∈D(O2), with ρ≡1inO3. Next, we put u(x, t)=p(t)ρ(x)Δϕ(x, T −t)∀(x, t)∈RN×(0,T),(3.37) where we have denoted by pthe function p(t)=sλ e−s βγ. From (3.10), we obtain (Δϕ(T−t))t−Δ(Δϕ(T−t)) = fin Q,(3.38) with f(t) = Δ((Dϕ)y)(T−t)−∇(∇·((Dϕ)y)(T−t)).(3.39) This provides the following heat problem for the function u: ut−Δu=Fin RN×(0,T), u(0) = 0 in RN, (3.40) with F(t)=pρf +pρΔϕ(T−t)−2p∇ρ·∇Δϕ(T−t)−pΔρΔϕ(T−t)(3.41) for all t∈(0,T). Then we rewrite Fin the form F=F1+F2, with F1(t)=pΔ(ρ((Dϕ)y)(T−t)) −p∇(∇·(ρ((Dϕ)y)(T−t))) +pΔ(ρϕ(T−t)) (3.42) and F2(t)=−2p∇ρ·∇((Dϕ)y)(T−t)−pΔρ((Dϕ)y)(T−t) +p∇(∇ρ·((Dϕ)y)(T−t)) + p∇ρ·(∇·((Dϕ)y)(T−t)) −2p∇ρ·∇ϕ(T−t)−pΔρϕ(T−t)−2p∇ρ·∇Δϕ(T−t) −pΔρΔϕ(T−t), (3.43) and we define ui(i=1,2) as the solution to ui t−Δui=Fiin RN×(0,T), ui(0) = 0 in RN. (3.44) An estimate of u1.Let us remark that F1∈L2(0,T;H−2(RN)N), while we would like to estimate the L2-norm of u1. Accordingly, we have to define u1like a solution by transposition. More precisely, u1is the unique function in L2(RN×(0,T))Nverifying RN×(0,T ) u1·kdxdt=RN×(0,T )pρ((Dϕ)y(T−t)) ·Δzdxdt −RN×(0,T )pρ((Dϕ)y(T−t)) ·∇(∇·z)dx dt +RN×(0,T )pρϕ(T−t)·Δzdxdt (3.45) Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1442 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO In conclusion, we get from (3.90) I(s, λ;ϕ)≤Cs 16λ40(1 + T2)O×(0,T ) e−8sα+6sα∗ ξ16 |ϕ|2dx dt(3.92) for s≥C(T4+T8) and λ≥C(1 + y∞+yt2 L2(Lσ)+eCTy2 ∞), which is exactly (3.13). 3.3. Some additional comments. In this subsection, we will indicate several recent results. The first two concern the controllability of the three-dimensional Stokes and Navier–Stokes systems with two scalar controls. The third one is related to the null controllability of the Stokes system with slip boundary conditions. 3.3.1. Three-dimensional Stokes and Navier–Stokes systems and two scalar controls. Let us consider system (3.7) with N= 3. What we would like to do now is to control acting over a (small) part of Ω just in two directions. More precisely, if we denote the control by v=(v1,v 2,v 3), we intend to drive the system (3.7) to zero at time t=Twith the restriction v1=0 or v2=0 or v3=0.(3.93) In the context of approximate controllability, this question was considered (and solved) in [49] for cylindrical domains Ω. We will have to impose an additional restriction on the control domain O, namely, ∃x0∈∂Ω,∃ε>0 such that O∩∂Ω⊃B(x0;ε)∩∂Ω.(3.94) The following result holds. Theorem 3.6. Let Tbe an arbitrary positive time. Let (3.9) and (3.94) be fulfilled. Then, for every y0∈H, there exists a two-dimensional L2control vsuch that the solution to (3.7) satisfies y(T)=0 in Ω.(3.95) Let us indicate briefly the idea of the proof. It is not restrictive to assume that B(x0;ε)∩Ω⊂O and, for instance, n3(x)=0 ∀x∈B(x0;ε)∩∂Ω. In view of the general arguments proved in section 1, the key point is to get a Carleman inequality of the form s−1Q ρ2|ϕ|2dx dt ≤CO×(0,T ) ρ2|ϕ1|2+|ϕ2|2dx dt (3.96) for the solutions to (3.10). Let us set O0=B(x0;ε)∩Ω. We start from (3.13), which must hold for some C>0. In O0,wehave ϕ3(x, t)=l(x,t) (∂1ϕ1+∂2ϕ2)(x1,x 2,ξ 3,t)dξ3,(3.97) where l(x, t) is an appropriate segment in O0that starts from (x, t), is parallel to e3, and ends at a point on ∂Ω. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1443 Using (3.97) in (3.13), in view of Fubini’s formula, H¨older’s inequality, and standard localization arguments, together with the particular definition of the set O0, one can find a Carleman inequality with local terms of |ϕ1|2and |ϕ2|2on the right-hand side. This and the dissipativity of system (3.10) imply, in a usual way, an observability inequality for the associated solutions with only these two terms on the right-hand side, which yields the result stated in Theorem 3.6. As a consequence of Theorem 3.6, following again the arguments developed in [37] and [24], one can deduce the local exact controllability to the trajectories of the Navier–Stokes system with two scalar controls: Theorem 3.7. Let (y,p)be a solution of (3.15) verifying (3.9) and y(0) ∈E= H∩L4(Ω)3. There exists ε>0such that, for each y0∈Esatisfying y0−y(0)L4≤ε, one can find a two-dimensional L2control vand an associated solution (y,p)to (3.14) for which we have y(T)=y(T)in Ω.(3.98) 3.3.2. Stokes systems with slip boundary conditions. Let us introduce the following system: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ yt−Δy+∇p=v1Oin Q, ∇·y=0 in Q, y·n=0,((Dy)·n)tg +(A(x, t)y)tg = 0 on Σ, y(0) = y0in Ω. (3.99) Here, we have used the notation atg to denote the tangential component of any vector a∈RN: atg =a−(a·n)n.(3.100) We will impose the following assumptions on A: A∈H(3−δ)/2(0,T;H1/2(∂Ω)N×N), A∈H1−δ(0,T;H−1+N/2+2δ(∂Ω)N×N), (3.101) where 0 <δ<1/2 arbitrarily close to 1/2 and δ<δ <1/2. Then we have the next result from [35]. Theorem 3.8. Let us assume that Ais a uniformly elliptic matrix verifying A∈L∞(Σ)N×Nand (3.101). Then, for each y0∈H, there exists v∈L2(O×(0,T))N such that the solution to (3.99) satisfies y(T)=0 in Ω.(3.102) Remark 10. It would be interesting to know if this result still holds, for general uniformly elliptic A∈L∞(Σ)N×N, without the hypotheses (3.101). However, this seems to be a difficult question. For similar systems for the Stokes or Navier–Stokes equations involving nonlinear slip boundary conditions, the situation is much more complicated. See [35] for more details. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1444 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO REFERENCES [1] V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin,Optimal Control, Contemp. Soviet Math., Consultants Bureau, New York, 1987. [2] S. Anita and V. Barbu,Null controllability of nonlinear convective heat equations, ESAIM Control Optim. Calc. Var., 5 (2000), pp. 157–173. [3] J.-P. Aubin,L’analyse non lin´eaire et ses motivations ´economiques, Masson, Paris, 1984. [4] S. A. Avdonin and S. A. Ivanov,Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, Cambridge, UK, 1995. [5] V. Barbu,Exact controllability of the superlinear heat equation, Appl. Math. Optim., 42 (2000), pp. 73–89. [6] V. Barbu,Controllability of parabolic and Navier-Stokes equations, Sci. Math. Jpn., 56 (2002), pp. 143–211. [7] V. Barbu and M. Iannelli,Controllability of the heat equation with memory, Differential Integral Equations, 13 (2003), pp. 1393–1412. [8] V. Barbu and G. Tessitore,Null controllability of stochastic heat equations with a multiplicative noise, Appl. Math. Optim., 47 (2000), pp. 97–120. [9] O. Bodart, M. Gonz´ alez-Burgos, and R. P´ erez-Garc´ ıa,Insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, C. R. Math. Acad. Sci. Paris, 335 (2002), pp. 677–682. [10] J.-M. Coron,On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions, ESAIM Control Optim. Calc. Var., 1 (1996), pp. 35–75. [11] J.-M. Coron and A. V. Fursikov,Global exact controllability of the 2-D Navier-Stokes equations on a manifold without boundary, Russian J. Math. Phys., 4 (1996), pp. 1–19. [12] L. de Teresa,Insensitizing controls for a semilinear heat equation, Comm. Partial Differential Equations, 25 (2000), pp. 39–72. [13] A. Doubova and E. Fern´ andez-Cara,Some control results for simplified one-dimensional models of fluid-solid interaction, Math. Models Methods Appl. Sci., 15 (2005), pp. 783–824. [14] A. Doubova, E. Fern´ andez-Cara, and M. Gonz´ alez-Burgos,Controllability results for linear viscoelastic fluids of the Maxwell and Jeffreys kinds, C. R. Acad. Sci. Paris, 331 (2000), pp. 537–542. [15] A. Doubova, E. Fern´ andez-Cara, and M. Gonz´ alez-Burgos,On the controllability of the heat equation with nonlinear boundary Fourier conditions, J. Differential Equations, 196 (2004), pp. 385–417. [16] A. Doubova, E. Fern´ andez-Cara, M. Gonz´ alez-Burgos, and E. Zuazua,On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim., 41 (2002), pp. 798–819. [17] Yu. V. Egorov,Some problems in the theory of optimal control, Z. Vycisl. Mat. i Mat. Fiz., 3 (1963), pp. 887–904 (in Russian). [18] C. Fabre,Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems, ESAIM Control Optim. Calc. Var., 1 (1995/96), pp. 267–302. [19] C. Fabre, J.-P. Puel, and E. Zuazua,Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), pp. 31–61. [20] E. Fern´ andez-Cara,Null controllability of the semilinear heat equation, ESAIM Control Optim. Calc. Var., 2 (1997), pp. 87–103. [21] E. Fern´ andez-Cara, M. Gonz´ alez-Burgos, S. Guerrero, and J.-P. Puel,Null controllability of the heat equation with Fourier Fourier boundary conditions: The linear case, ESAIM Control Optim. Calc. Var., 12 (2006), pp. 442–465. [22] E. Fern´ andez-Cara, M. Gonz´ alez-Burgos, S. Guerrero, and J.-P. Puel,Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: The semilinear case, ESAIM Control Optim. Calc. Var., 12 (2006), pp. 466–483. [23] E. Fern´ andez-Cara and S. Guerrero,Global Carleman estimates for solutions of parabolic systems defined by transposition and some applications to controllability,AMRX Appl. Math. Res. Express, Volume 2006 (2006), article 75090; available online from http://www.hindawi.com/GetArticle.aspx?doi=10.1155/AMRX/2006/75090. [24] E. Fern´ andez-Cara, S. Guerrero, O. Yu. Imanuvilov, and J.-P. Puel,Local exact controllability to the trajectories of the Navier-Stokes equations, J. Math. Pures Appl., 83 (2004), pp. 1501–1542. [25] E. Fern´ andez-Cara, S. Guerrero, O. Yu. Imanuvilov, and J.-P. Puel,Some controllability results for the N-dimensional Navier–Stokes and Boussinesq systems with N−1scalar controls, SIAM J. Control Optim., 45 (2006), pp. 146–173. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CARLEMAN, PARABOLICITY, AND CONTROLLABILITY 1445 [26] E. Fern´ andez-Cara and L. de Teresa,Null controllability of a cascade system of parabolichyperbolic equations, Discrete Contin. Dyn. Syst., 11 (2004), pp. 699–714. [27] E. Fern´ andez-Cara and E. Zuazua,The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5 (2000), pp. 465–514. [28] E. Fern´ andez-Cara and E. Zuazua,Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 17 (2000), pp. 583–616. [29] A. V. Fursikov and O. Yu. Imanuvilov,Local exact controllability of the Navier-Stokes equations, C. R. Acad. Sci. Paris S´er. I Math., 323 (1996), pp. 275–280. [30] A. V. Fursikov and O. Yu. Imanuvilov,Controllability of Evolution Equations, Lecture Note Series 34, Research Institute of Mathematics, Seoul National University, Seoul, 1996. [31] A. V. Fursikov and O. Yu. Imanuvilov,Exact controllability of the Navier-Stokes and Boussinesq equations, Uspekhi Mat. Nauk, 54 (1999), pp. 93–146 (in Russian); English translation in Russian Math. Surveys, 54 (1999), pp. 565–618. [32] Y. Giga and H. Sohr,Abstract Lpestimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), pp. 72–94. [33] O. Glass,An addendum to a J. M. Coron theorem concerning the controllability of the Euler system for 2D incompressible inviscid fluids, J. Math. Pures Appl. (9), 80 (2001), pp. 845– 877. [34] R. Glowinski and J.-L. Lions,Exact and approximate controllability for distributed parameter systems, in Acta Numerica, Cambridge University Press, Cambridge, UK, 1995, pp. 159– 333. [35] S. Guerrero,Local exact controllability to the trajectories of the Navier-Stokes system with nonlinear Navier-slip boundary conditions, ESAIM Control Optim. Calc. Var., 12 (2006), pp. 484–544. [36] O. Yu. Imanuvilov,Controllability of parabolic equations, Sb. Math., 186 (1995), pp. 109–132 (in Russian). [37] O. Yu. Imanuvilov,Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), pp. 39–72. [38] O. Yu. Imanuvilov and J.-P. Puel,Global Carleman estimate for weak elliptic non homogeneous Dirichlet problem, Int. Math. Res. Not., 16 (2003), pp. 883–913. [39] O. Yu. Imanuvilov and M. Yamamoto,Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, in Control of Nonlinear Distributed Parameter Systems, Lecture Notes in Pure and Appl. Math. 218, Marcel Dekker, New York, 2001, pp. 113–137. [40] O. A. Ladyzhenskaja, V. A. Solonnikov, and N. N. Ural’ceva,Linear and Quasilinear Equations of Parabolic Type, translated from the Russian by S. Smith, Translations of Mathematical Monographs 23, AMS, Providence, RI, 1967. [41] G. Lebeau and L. Robbiano,Contrˆole exact de l’´equation de la chaleur, Comm. Partial Differential Equations, 20 (1995), pp. 335–356. [42] G. Lebeau and E. Zuazua,Null-controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal., 141 (1998), pp. 297–329. [43] G. Lebeau and E. Zuazua,Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Rational Mech. Anal., 148 (1999), pp. 179–231. [44] J.-L. Lions,Exact controllability, stabilizability and perturbations for distributed systems, SIAM Rev., 30 (1988), pp. 1–68. [45] J.-L. Lions,Contrˆolabilit´e Exacte, Perturbations et Stabilisation de Syst`emes Distribu´es, Tome 1, Rech. Math. Appl. 8, Masson, Paris, 1988. [46] J.-L. Lions,Contrˆolabilit´e Exacte, Perturbations et Stabilisation de Syst`emes Distribu´ es, Tome 2, Rech. Math. Appl. 9, Masson, Paris, 1988. [47] J.-L. Lions,Quelques notions dans l’analyse et le contrˆole de syst`emes `a donn´ees incompl`etes, in Proceedings of the 11th Congress on Differential Equations and Applications/First Congress on Applied Mathematics, University of M´alaga, M´alaga, 1989, pp. 43–54. [48] J.-L. Lions,Are there connections between turbulence and controllability?, in Analysis and Optimization of Systems, A. Bensoussan and J.-L. Lions, eds., Lecture Notes Control and Inform. Sci. 144, Springer-Verlag, Berlin, 1990. [49] J.-L. Lions and E. Zuazua,A generic uniqueness result for the Stokes system and its control theoretical consequences, in Partial Differential Equations and Applications, P. Marcellini, G. Talenti, and E. Visentini, eds., Lecture Notes in Pure and Appl. Math. 177, Marcel Dekker, New York, 1996, pp. 221–235. [50] J.-L. Lions and E. Zuazua,Exact boundary controllability of Galerkin’s approximations of Navier-Stokes equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), pp. 605–621. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1446 ENRIQUE FERN´ ANDEZ-CARA AND SERGIO GUERRERO [51] J.-P. Puel,Controllability of Partial Differential Equations, Lectures in the Universidade Federal do Rio de Janeiro, Brazil, 2002. [52] D. L. Russell,A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Int. Math. Res. Not., 52 (1973), pp. 189–221. [53] D. L. Russell,Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), pp. 639–739. [54] Th. I. Seidman,On uniform null-controllability and blow-up estimates, in Control Theory of Partial Differential Equations, Lect. Notes Pure Appl. Math. 242, Chapman & Hall/CRC, Boca Raton, FL, 2005, pp. 213–227. [55] R. Temam,Navier-Stokes Equations. Theory and Numerical Analysis, 2nd ed., North-Holland, Amsterdam, 1977. [56] E. Zuazua,Exact boundary controllability for the semilinear wave equation, in Nonlinear Partial Differential Equations and Their Applications, Vol. 10, Pitman Res. Notes Math. Ser. 220, Longman, Harlow, UK, 1991, pp. 357–391. [57] E. Zuazua,Controllability of the linear system of thermoelasticity, J. Math. Pures Appl., 74 (1995), 303–346. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php